An inverse eigenvalue problem of the wave equation for a multi-connected region in R2 together with three different types of boundary conditions

Abstract

The trace of the wave kernel μ(t)=∑ ∞ ω=1 exp(− i tE 1/2 ω) , where { E ω } ω=1 ∞ are the eigenvalues of the negative Laplacian − ∇ 2=−∑ 2 k=1( ∂ ∂x k ) 2 in the ( x 1, x 2)-plane, is studied for a variety of bounded domains, where −∞< t<∞ and i = −1 . The dependence of μ( t) on the connectivity of bounded domains and the Dirichlet, Neumann and Robin boundary conditions are analyzed. Particular attention is given for a multi-connected vibrating membrane in R 2 surrounded by simply connected bounded domains Ω j with smooth boundaries ∂Ω j (j=1,…,n) , where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Γ i (i=1+k j−1,…,k j) of the boundaries ∂Ω j is considered, such that ∂Ω j=⋃ k j i=1+k j−1 Γ i where k 0=0. The basic problem is to extract information on the geometry of Ω using the wave equation approach from complete knowledge of its eigenvalues. Some geometrical quantities of Ω (e.g. the area of Ω, the total lengths of its boundary, the curvature of its boundary, the number of the holes of Ω, etc.) are determined from the asymptotic expansion of the trace of the wave kernel μ( t) for small | t|.